text: The perturbation series are in terms of a small ordering parameter  – which subsequently turns out to be proportional to (and of the order of) the wave slope ka, see the series solution in this section. So, take :
  When applied in the flow equations, they should be valid independent of the particular value of ε. By equating in powers of ε, each term proportional to ε to a certain power has to equal to zero. As an example of how the perturbation-series approach works, consider the non-linear boundary condition ; it becomes:
  The resulting boundary conditions at  for the first three orders are:
  ;First order:
  ;Second order:
  ;Third order:
  In a similar fashion – from the dynamic boundary condition  – the conditions at  at the orders 1, 2 and 3 become:
  ;First order:
  ;Second order:
  ;Third order:
  For the linear equations ,   and  the perturbation technique results in a series of equations independent of the perturbation solutions at other orders:
  The above perturbation equations can be solved sequentially, i.e. starting with first order, thereafter continuing with the second order, third order, etc.